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Topic: Non standard probability theory
Replies: 12   Last Post: Dec 19, 2013 8:03 PM

 Messages: [ Previous | Next ]
 ross.finlayson@gmail.com Posts: 2,720 Registered: 2/15/09
Re: Non standard probability theory
Posted: Dec 12, 2012 12:19 AM

On Dec 11, 3:26 pm, FredJeffries <fredjeffr...@gmail.com> wrote:
> On Nov 11, 6:36 pm, kir <danielgold...@sunyorange.edu> wrote:
>
>
>

> > What's really true is that it occurs with infinitesimal probability.
>
> Alexander R Prusshttp://alexanderpruss.blogspot.com/
> has written on this subject
>
> "Infinite Lotteries, Perfectly Thin Darts and Infinitesimals"https://bearspace.baylor.edu/Alexander_Pruss/www/papers/Lotteries.pdf
>
> "Probability, Regularity and Cardinality"https://bearspace.baylor.edu/Alexander_Pruss/www/papers/Regularity.pdf

Consider the real numbers of the unit interval as the Cantor space
{0,1}^N. Then to sample one with a fair coin toss or Bernoulli event,
that is the initial segment of the sample, but the next sample is both
a refinement of the first, and a sample of its own. Ad infinitum:
taking one sample from the Cantor space, using the simplest tool of
the symmetrical binary decision, takes infinitely many samples from
the Cantor space.

Then there's the consideration of which real number is the result. If
the first sample is an irrational, which may seem expected due the
truly random nature of the fair coin toss, infinitely many different
irrationals thus result as the sample. However, in the event when a
rational number is sampled, the repeating part of the expansion's
representation thus results as a sample infinitely many times. The
sample that is the rational is infinitely less likely to occur, but
when it does, it occurs infinitely many times over.

It's quite interesting to read of these developments in the progress
of probability theory. Thanks again, Fred.

Regards,

Ross Finlayson